Let $b,c,$ and $m$ be positive integers and suppose $a$ is relatively prime to $m$. Furthermore, assume $a^b\equiv a^c\equiv 1\pmod{m}$ and that $g=\gcd(b,c)$.
I know that since $g=\gcd(b,c)$ it follows that $g|b$ and $g|c$. So one could write $b=gx$ and $c=gy$ for some $x,\;y\;\epsilon\;\mathbb Z$. I then have $a^b\equiv a^{gx}\equiv (a^{g})^x\equiv 1\pmod{m}$. I don't quite understand how to prove $a^g\equiv 1\pmod{m}$. I'm wondering if I should be utilizing the fact that the order of $a$ divides any power, $k$, such that $a^k\equiv 1\pmod{m}$. Or if I was on the right track just need to manipulate $b$ and $c$ slightly differently. Any help would be much appreciated.